Fourier Series Visualizer

Joseph Fourier introduced the decomposition of periodic functions into infinite sums of sines and cosines in his 1807 manuscript on heat conduction, later published in the landmark 1822 treatise Théorie analytique de la chaleur. The mathematical establishment of the day was deeply skeptical — Lagrange himself criticized Fourier\'s claim that any periodic function, even one with discontinuities, could be represented this way — but by the 1830s Dirichlet and others had proven the necessary convergence conditions, and Fourier analysis became a cornerstone of mathematical physics.

The Gibbs phenomenon — the ~9% overshoot near discontinuities in partial-sum Fourier series — was first observed experimentally by Albert Michelson around 1898 while building a mechanical harmonic analyzer. He was convinced his instrument was faulty until J. Willard Gibbs proved theoretically in 1899 that the overshoot is an intrinsic property of Fourier series, not a measurement error. It persists regardless of how many harmonics you add — only the width of the overshoot narrows, not its height.

Today, Fourier analysis is essential to virtually every engineering discipline: audio synthesis (additive synthesizers literally implement Fourier series in real time), EMI analysis (square-wave clocks radiate at every odd harmonic), image compression (JPEG uses DCT, a Fourier cousin), MRI reconstruction, quantum mechanics (momentum-space wavefunctions), and signal-processing theory. This visualizer demonstrates the core principle that took 100 years to gain full mathematical acceptance.

Pick square, triangle, or sawtooth and move the harmonic-count slider. The visualizer draws the partial sum alongside the ideal waveform, showing both convergence and the Gibbs overshoot at discontinuities. Accompanying readouts report harmonic type (odd-only for square/triangle; all for sawtooth) and amplitude decay rate (1/n for square/sawtooth; 1/n² for triangle, hence faster convergence).

Useful for: understanding the harmonic content of digital clock signals (EMI analysis), seeing why triangle waves sound smoother than squares (faster spectral decay), and building intuition for how windowing reduces spectral leakage. Everything runs client-side; no values leave your browser.